Apparatus of the type referred to above is well known. See for example "Applications of Optimal Control Concepts to Digital Shaker Control Systems" M. A. Underwood, Journal of the Society of Environmental Engineers, December 1981 pages 19 to 22. The apparatus is used in the testing of a variety of systems, for example equipment which will be subject to vibration in a vehicle or a model of an engineering structure for an area liable to earthquake, to mention just two of many possibilities. The reference power spectrum is determined in accordance with the properties of the real life drive system, e.g. by recording the frequency spectrum of a vehicle driven over rough terrain in the case of the first example above. The feedback loop is employed because the driver used in the apparatus cannot act as an infinite energy source and the actual vibrations differ from the drive signal in dependence upon the transfer function of the system under test. It is the power spectrum of the actual vibrations which has to correspond to the reference spectrum because, in real life the drive system does act substantially as an infinite energy source.
The phase randomization is known to be desirable for various reasons, including ensuring a Gaussian amplitude distribution and rendering the digital technique comparable with a purely analog technique. It is known to effect the randomization by associating with the set of amplitude values at the input to the IDFT means a set of random numbers which determine the phases of the corresponding frequency components in the time domain drive signal at the output of the IDFT means. The amplitude values and random numbers provide R,.crclbar. inputs to the IDFT.
The basic form of the known apparatus is illustrated in FIG. 1 of the drawings. The drive signal x.sub.n (t) is applied via an amplifier 11 to the driver 10 which drives the system under test 12 (SUT) and which may be one of various commercially available devices such as shaker table or a powerful loudspeaker. The driver and SUT together have a transfer function H(w) in the frequency domain. A transducer 14, again a commercially available device, provides the actual vibration signal y.sub.n (t) which is applied via an amplifier 15 to the DFT means 16 whose outputs is Y.sub.n (w). This data and the reference spectrum Z(w) from a memory device 18 are applied to means 20 effecting the equalization algorithm, which is explained below. The resulting new drive spectrum X.sub.n+1 (w) and the random number set R.sub.n (w) form a generator 22 are applied to the IDFT means 24 which provide x(t).
The function of the equalization algorithm means 20 can be explained as follows.
The system is equalized when EQU Y(w)=X(w)*H(w) O Z(w) or X(w)=Z(w)/H(w) (1)
Since x(t) is a random process a single response spectrum Y.sub.n (w) gives random values of power spectral density. A true power spectral density estimate can only be derived by averaging many instantaneous results. ##EQU1## The digital implementation of this control loop is effected by generating sample blocks of data x.sub.n (t) from estimated drive power spectra X.sub.n (w). The resulting system response y.sub.n (t) is sampled and fourier transformed to produce a response spectrum Y.sub.n (w).
From equation (1) it can be seen that Y(w) can be equalized using the iteration ##EQU2## However this relies on Y(w) being a sufficiently accurate estimate of the response power spectral density.
Two methods of averaging are commonly used to reduce variance in Y(w). Exponential averaging allows frequent small changes to be made in drive spectrum. However loop stability is poor for the averaging time constants required to reduce variance sufficiently. This results in large over and under shoots in the estimated drive level and hence the overall equalization time is high.
Fixed memory averaging relies on driving an accurate stimate of Y(w) before any equalization is done. This means that the drive is changed by large amounts infrequently. For a linear SUT a more stable loop is produced but equalization is slow because of the large averaging time required. Moreover loop instability is more likely to be caused by non-linear elements in the SUT.